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# Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

• * Corresponding author: Phanuel Mariano

† Research was supported in part by NSF Grant DMS-1262929.
‡ Research was supported in part by NSF Grant DMS-1405169, 1712427.
** Research was supported in part by the UConn Mathematics Department and a Zuckerman fellowship.

• We consider three matrix models of order 2 with one random entry $\epsilon$ and the other three entries being deterministic. In the first model, we let $\epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right)$. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $\epsilon\sim \rm{Bernoulli}\left(p\right)$ and $p\in [0, 1]$ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

Mathematics Subject Classification: Primary: 37H15; Secondary: 37M25, 60B20, 60B15, 11B39.

 Citation: • • Figure 1.  Histogram

Figure 3.  $n = 1\, 000\, 000$

Figure 4.  $\lambda(\xi)$ vs. $\xi$

Figure 5.  $k = 0.01$, $n = 1000$, $m = 1\, 000\, 000$

Figure 6.  $k = 0.25$, $n = 1000$, $m = 5\, 000\, 000$

Figure 7.  $k = 0.01$, $n = 1000$, $m = 1\, 000\, 000$

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